Are Infinite Points on a Straight Line Just One Point?

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Discussion Overview

The discussion revolves around the conceptual understanding of a straight line in relation to the infinite number of points it contains. Participants explore the implications of dividing a line into smaller segments and whether such divisions can lead to the conclusion that a line is essentially a single point. The conversation touches on mathematical definitions, visual perceptions, and philosophical interpretations of points and lines.

Discussion Character

  • Exploratory
  • Conceptual clarification
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant suggests that if a straight line consists of an infinite number of points, repeatedly dividing it should not lead to the conclusion that it becomes a single point.
  • Another participant argues that a line requires at least two points to define it, implying that a single point cannot represent a line.
  • Some participants propose that while a line can be infinitely divided, it will always retain a length and never actually become a point.
  • There is a discussion about the nature of points, with one participant asserting that a point is not merely a dot, but a distinct concept that encompasses an infinite number of values.
  • Another participant questions the meaning of "connected" in the context of infinite points and suggests that the visual representation of a line may not accurately reflect its mathematical properties.
  • One participant illustrates the concept of infinite division using a specific example of a line segment, emphasizing that it always has endpoints and an infinite number of points between them.

Areas of Agreement / Disagreement

Participants express differing views on whether a line can be considered a point, with no consensus reached. Some argue for the distinct nature of lines and points, while others challenge these definitions, leading to an unresolved discussion.

Contextual Notes

Participants highlight the difference between mathematical analysis and physical representation, noting that the definitions and interpretations of points and lines may vary based on context. The discussion also reveals uncertainties regarding the concept of infinity and the implications of dividing infinite sets.

  • #31
Werg22 said:
d(x,x)=0, what's the point of bringing this up? No one contested that.

You're saying it doesn't make sense to measure a point. That is a one right there.
 
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  • #32
nine said:
f(0) = 5
f(0+s) = 5
where s a very small fraction

No f(0+s) = 2s+5 and if s =/= 0 then f(0+s) =/= 5 no matter how small s is.

As s tends to 0 then f(0+s) tends to 5 but is only =5 when s=0

Look at JasonRox's example for 1/2^n n an integer

as n gets larger 1/2^n tends to 0 but no matter how large n is 1/2^n will still be none zero
 
  • #33
JasonRox said:
Points have no length, so it is 0 length each point.

So, this doesn't make sense to you:

0+0+0+0+...?

The answer is clearly 0.

The sum of a finite number of 0s is 0. In fact, even the sum of a countable number of 0s is 0. But a line contains an uncountable the sum of an uncountable number of 0s is not 0- in fact, it's not defined.
 
  • #34
JasonRox said:
You're saying it doesn't make sense to measure a point. That is a one right there.

Let me clarify: it doesn't make sense to get the measure of a point for the purpose of using it as a part of a distance.
 
  • #35
HallsofIvy said:
The sum of a finite number of 0s is 0. In fact, even the sum of a countable number of 0s is 0. But a line contains an uncountable the sum of an uncountable number of 0s is not 0- in fact, it's not defined.

That's what I talked about. I was just talking about that case.
 
  • #36
JasonRox said:
This is where it gets tricky. I'm not 100% sure how to answer it, but logic fails here though.

Because what you're saying here is that you're counting all the zeroes and it adds to zero. The truth is that you can't even count all the zeroes! There are uncountably many points in a line, so you can't count them all even though they're all zero.

That's the post I'm talking about HallsofIvy.

The quote you posted has to do with werg22 talking about counting infinite many lengths not making sense to me, and so I commented...

Adding, 0+0+0+0+0+...=0, does not make sense?

That is infinitely many lengths in which in fact does make sense to add because there are countably many.
 

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